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The real Seifert form and the spectral pairs of isolated hypersurface singularities. (English) Zbl 0851.14015
For a hypersurface singularity $$f: (\mathbb{C}^{n+1}, 0)\to (\mathbb{C}, 0)$$, a discrete invariant $$\text{Spp} (f)$$ called the set of spectral pairs is defined as an element in the free abelian group generated by $$\mathbb{Q} \times \mathbb{N}$$. It determines and is determined by the collection of Hodge numbers of the cohomology of the Milnor fiber of $$f$$. The main result of this article stated in the beginning part is the following:
Theorem. Consider the image $$\text{Spp}_{\text{mod-}2} (f)$$ of $$\text{Spp} (f)$$ under the projection induced by $\mathbb{Q} \times \mathbb{N}\to (\mathbb{Q}/ 2\mathbb{Z}) \times \mathbb{N}.$ Then the information contained in $$\text{Spp}_{\text{mod-}2} (f)$$ is equivalent to the information contained in the real Seifert form of the singularity $$f$$.
Needless to say, this theorem has no meaning unless the meaning of the phrase “the information contained in the real Seifert form of the singularity $$f$$” is made clear. Let $$F$$ denote the Milnor fiber of $$f$$ and let $$U= \widetilde {H}_n (F, \mathbb{R})$$. The monodromy defines an isomorphism $$h: U\to U$$ and the intersection form defines a map $$b:U\to U^*= \operatorname{Hom} (U, \mathbb{R})$$. Besides, in case $$n>0$$ we have the canonical identification $$U^*= H_n (F, \partial F, \mathbb{R})$$ and the variation isomorphism $$\text{Var}: U^*\to U$$ is defined. The real Seifert form $$L$$ is the bilinear form on $$U$$ defined by $$L(a, b)= \langle \text{Var}^{-1} (a), b\rangle$$ for $$a, b\in U$$, where $$\langle\;, \;\rangle: U^* \times U\to \mathbb{R}$$ denotes the canonical pairing. The author defines two bilinear forms on real vector spaces $$U_1$$ and $$U_2$$ to be isomorphic, if their induced hermitian forms on $$U_1 \otimes \mathbb{C}$$ and $$U_2 \otimes \mathbb{C}$$ are isomorphic in a natural manner.
The above theorem implies that $$\text{Spp}_{\text{mod-} 2} (f)$$ determines and is determined by the isomorphism class of the linear form $$L$$ on $$U$$. In order to verify the theorem the author considers the quadruplet $$(U \otimes \mathbb{C}, b\otimes 1_\mathbb{C}, h\otimes 1_\mathbb{C},\text{Var} \otimes 1_\mathbb{C})$$ instead of the Seifert form $$L$$. This quadruple is the model of the generalized concept called hermitian variation structures. The theory of hermitian variation structures is developed in this article.
Reviewer: T.Urabe (Tokyo)

##### MSC:
 14J17 Singularities of surfaces or higher-dimensional varieties 32S55 Milnor fibration; relations with knot theory 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
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